Statistics

Statistics

1. Type of data

   • Quantitative
     • Type of data
       • Continuous
         • It takes any values continuously defined within a specific range
       • Discrete
         • It takes particular values. It can ben infinite or finite number.
     • Usually represented by:
       • Cumulative percentage plot
         • It shows for any value what proportion of observations were less than that value.
       • Stem-and-leaf plot
         • Individual values are retained
       • Histogram
   • Qualitative
     • Include characteristics and description. *Cannot be numerical
     • Usually represented as bar graph or pie chart

2. Fundamental requirements

   • Any sampling experiment must contain representative samples and successive observations must be independent

3. Terminology to describe what is observed in an experiment

   • Sample space (S)
     • Set all of possible outcomes of an experiment
   • Event
     • Any subset of S
     • An event is said to occur if it contains the actual outcome

4. Axioms of probability

5. Random variables

   • S of possible numerical values together with a probability over those values
   • Discrete
     • The behaviour can be represented graphically by means of a
       • Distribution function
         • Based on the set of point in S and the probability of this event
       • Probability mass function
   • Continuous
     • The behaviour is described by
       • Density function fx(x)
         • It has to be understood in terms of the
           • Distribution function Fx(x)
             • Measure the probability that the value of random variable is less than or equal to the argument x
             • In this case, because there're no discrete steps in probability, Fx(x) is continuous and differentiable, and its derivative is called
               • Probability density function fx(x)
                 • f indicates the concentration of possible observed values along the real axis
   • Notation: Capital letter for the random value and lower-case for the observe value
   • Properties of density and distribution functions

6. Measure of location and dispersion

   • It is assumed that the random variable is one of a class whose distribution is specified by a formula
     • Binomial
     • Normal
     • Poisson
   • Characterize the random variable in terms of two numbers. In practice both are used together
     • Location
       • 'Typical' value
       • Three common measures
         • Mean
           • Represented as u_x
         • Median
           • Continuous random variable
           • Discrete random variable
           • The median of a distribution does not coincide with mean unless the distribution has an axis of symmetry
         • Mode
           • It's any point for which the probability density function P_x(mode)(if discrete) or the probability density function f_x(mode)(if continuous) is an overall maximum.
         • Two approaches to measure the variation of random variables around their central values
           • Variance
             • A weighted sum of squared diff. between the possible values and the mean. Written as Var(X) or sigma^2*x
               • It measures how tightly concentrated the possible values are about the mean.
             • In order to have the result without square unit produce by variance. It is used:
               • Standard deviation sigma_x
                 • Represented as the square root of the variance
           • Quartiles
             • Exploit the distribution function
               • Divide the range of possible values of X into four succesive intervals, for each of which the probability of X falling in the interval is one-quarter
     • Dispersion
       • 'Spread' about that value

7. Expected values

   • The mean and variance are special cases of expected values for a random variable.
     • The expected value of a function h(X) of X is:
       • An alternative expression of the variance:

8. Independent variable

   • When X and Y are independent the so-called joint probabilities can be found from the formula:
   • Properties of the sum of two random variables
   • Adding a constant to a random variable
   • Multiplying a constant to a random variable

9. Measures of sample data

   • Sample average
   • Sample variance
   • Sample standard deviation
   • The distribution of the quantity under investigation is usually not known
     • We need the mean and variance(standard deviation0 to characterize the data)
   • Variance of sample average
     • The sample average /X itself is a random variable (whereas the observed sample mean /x is a number)
       • *n: number of data in the sample
   • Sample median
     • provides an estimate of the true median
   • For small samples a common measure of dispersion is
     • Sample range
       • diff between the largest and smallest elements of the data set

10. Binomial distribution

    • Bernoulli trial
      • Features
        • The outcome can be classified as either a success or a failure
        • The probability of success is p and the probability of failure is 1-p
      • Bernoulli random variable X is a variable assigned to represent the successes in a Bernoulli trial

11. Poisson distribution

    • Distribution without upper limit
    • X is the number of times an event happens
    • x is said to have Poisson distribution with parameter λ and density function
    • it can be shown that
    • For a large "n" and small "p" the Poisson distribution approximates the binomial with λ=np
      • This approx. can be used if

12. The normal distribution

    • A continuous random variable X has a normal distribution with mean micro_X and standard deviation sigma_x its density function is
    • The graph of the density function is symmetrical about micro_x
    • Distribution function
    • Traditional solution by use of table giving values of the (cumulative) distribution function. *pg1078
    • Variance
    • Any normal random variable X can be considered to have been obtained from a standard normal random variable Z
    • The standard normal cumulative distribution function is

13. The central limit theorem

    • With sufficiently large random samples it's possible to compute:
      • Mean of the sample means
      • Standard deviation of the sample means
    • The sample means will be approx. normally distributed, provided the sample size is sufficiently large (usually n>25/30) and :

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